modular_forms.rst

.. index:: modular forms

*************
Modular forms
*************

One of SageMath's computational specialities is (the very technical field
of) modular forms and can do a lot more than is even suggested in
this very brief introduction.

Cusp forms
==========

How do you compute the dimension of a space of cusp forms using Sage?

To compute the dimension of the space of cusp forms for Gamma use
the command ``dimension_cusp_forms``. Here is an example from
section "Modular forms" in the Tutorial:

::

    sage: from sage.modular.dims import dimension_cusp_forms
    sage: dimension_cusp_forms(Gamma0(11),2)
    1
    sage: dimension_cusp_forms(Gamma0(1),12)
    1
    sage: dimension_cusp_forms(Gamma1(389),2)
    6112

Related commands: ``dimension_new__cusp_forms_gamma0`` (for
dimensions of newforms), ``dimension_modular_forms`` (for modular
forms), and ``dimension_eis`` (for Eisenstein series). The syntax is
similar - see the Reference Manual for examples.

.. index:: cosets of Gamma_0

Coset representatives
=====================

The explicit representation of fundamental domains of arithmetic
quotients :math:`H/\Gamma` can be determined from the cosets of
:math:`\Gamma` in :math:`SL_2(\ZZ)`. How are these cosets
computed in Sage?

Here is an example of computing the coset representatives of
:math:`SL_2(\ZZ)/\Gamma_0(11)`:

::

    sage: G = Gamma0(11); G
    Congruence Subgroup Gamma0(11)
    sage: list(G.coset_reps())
    [
    [1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]
    [0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [ 1  4], [ 1  5], [ 1  6],
    <BLANKLINE>
    [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]
    [ 1  7], [ 1  8], [ 1  9], [ 1 10]
    ]


.. index:: modular symbols, Hecke operators

Modular symbols and Hecke operators
===================================

Next we illustrate computation of Hecke operators on a space of
modular symbols of level 1 and weight 12.

::

    sage: M = ModularSymbols(1,12)
    sage: M.basis()
    ([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
    sage: t2 = M.T(2)
    sage: f = t2.charpoly('x'); f
    x^3 - 2001*x^2 - 97776*x - 1180224
    sage: factor(f)
    (x - 2049) * (x + 24)^2
    sage: M.T(11).charpoly('x').factor()
    (x - 285311670612) * (x - 534612)^2

Here ``t2`` represents the Hecke operator :math:`T_2` on the space
of Full Modular Symbols for :math:`\Gamma_0(1)` of weight
:math:`12` with sign :math:`0` and dimension :math:`3` over
:math:`\QQ`.

::

    sage: M = ModularSymbols(Gamma1(6),3,sign=0)
    sage: M
    Modular Symbols space of dimension 4 for Gamma_1(6) of weight 3 with sign 0
    over Rational Field
    sage: M.basis()
    ([X,(0,5)], [X,(3,5)], [X,(4,5)], [X,(5,5)])
    sage: M._compute_hecke_matrix_prime(2).charpoly()
    x^4 - 17*x^2 + 16
    sage: M.integral_structure()
    Free module of degree 4 and rank 4 over Integer Ring
    Echelon basis matrix:
    [1 0 0 0]
    [0 1 0 0]
    [0 0 1 0]
    [0 0 0 1]

See the section on modular forms in the Tutorial or the Reference
Manual for more examples.

Genus formulas
==============

Sage can compute the genus of :math:`X_0(N)`, :math:`X_1(N)`,
and related curves. Here are some examples of the syntax:

::

    sage: from sage.modular.dims import dimension_cusp_forms
    sage: dimension_cusp_forms(Gamma0(22))
    2
    sage: dimension_cusp_forms(Gamma0(30))
    3
    sage: dimension_cusp_forms(Gamma1(30))
    9

See the code for computing dimensions of spaces of modular forms
(in ``sage/modular/dims.py``) or the paper by Oesterlé and Cohen {CO}
for some details.